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At time of writing the first definition of a $ (p, q) $-tensor on the Wikipedia page is as follows.


Definition. A $ (p, q) $-tensor is an assignment of a multidimensional array $$ T^{i_1\dots i_p}_{j_{1}\dots j_{q}}[\mathbf{f}] $$ to each basis $\mathbf{f}$ of an $n$-dimensional vector space such that, if we apply the change of basis $\mathbf{f}\mapsto \mathbf{f}\cdot R $ then the multidimensional array obeys the transformation law $$ T^{i'_1\dots i'_p}_{j'_1\dots j'_q}[\mathbf{f} \cdot R] = \left(R^{-1}\right)^{i'_1}_{i_1} \cdots \left(R^{-1}\right)^{i'_p}_{i_p} T^{i_1, \ldots, i_p}_{j_1, \ldots, j_q}[\mathbf{f}] R^{j_1}_{j'_1}\cdots R^{j_q}_{j'_q} . $$


This is a standard definition I can remember reading in textbooks during my undergraduate degree. To me, it also seems far too confusing. To understand a $ (p, q) $-tensor as an element in $$ \text{Hom}(\underbrace{V^* \otimes\dots\otimes V^*}_{p\text{}} \otimes \underbrace{V \otimes\dots\otimes V}_{q \text{}}, \mathbb{K}) $$ one only has to understand the tensor product on vector spaces (which is easy to define in terms of bases). To then recover the description of a multidimensional array one also has understand cobases, however these can also be easily explained constructively.

Question

Why would anyone give the standard definition?

I initially thought the answer lay in applied mathematics. However linear maps are omnipresent in applied mathematics and I have never seen a linear map defined as a function on bases that satisfies coherence with respect to base change. Furthermore I feel the consensus would be that this is a bad definition from a pedagogical point of view (I certainly think it is). So why is the analogous definition of $ (p, q) $-tensors standard?

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    $\begingroup$ Physicists and Computer Scientists are why $\endgroup$ – JJJ Mar 29 '19 at 19:14
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    $\begingroup$ When you do long computations in coordinates, operations like contractions and similar become automatic by using upper and lower indices. Or, at least, I have been told so :) $\endgroup$ – Francesco Polizzi Mar 29 '19 at 19:16
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    $\begingroup$ More seriously, the notion of "dual vector space" is often a delicate one, at least for non-mathematicians. Computations in coordinates can be less transparent than coordinate-free ones, but they can be grasped more easily by people whose background in abstract linear algebra is not strong. $\endgroup$ – Francesco Polizzi Mar 29 '19 at 19:21
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    $\begingroup$ I still like the definition I learned from my undergraduate differential geometry / linear algebra professor. "A vector is like a cabbage. A linear functional is like a goat, it eats a cabbage and spits out a number. [A goat with an unusual digestive system, I guess]. A $(p,q)$ tensor is like a dragon that eats $p$ goats and $q$ cabbages, and spits out a number." $\endgroup$ – Nate Eldredge Mar 30 '19 at 3:23
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    $\begingroup$ @NateEldredge: That cabbage/goat mnemonic only works if you remember that a cabbage can also eat a goat. $\endgroup$ – Steven Landsburg Mar 31 '19 at 4:18
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I think that the answer lies in the "educational culture" of physicists. Physicists are often used -well at least at the undergraduate level- to learn and perform complicated computations with abstract objects, without caring much about the structure and the abstract properties of the ambient spaces containing these abstract objects.
The definition of tensors as "generalized" vectors or matrices, with covariant and contravariant components, is one of such examples: Starting from such a definition, enables one to quickly learn how to perform computations with tensors, without demanding a deep understanding of the abstract definition of tensor products let alone dual spaces, manifolds, tangent and cotangent spaces or bundles etc. In this way, an undergraduate physicist quickly becomes able to perform computations in a wide range of topics (from classical mechanics to special and general relativity and from continuoum mechanics to electromagnetism and even field theories) while in most cases (s)he misses a deeper understanding of where all these objects "live".

Furthermore, this "educational culture" seems to be supported by the fact that the definition of tensors via their transformation properties usually arises -in physics texts- through phenomenological or semi-phenomenological considerations: For example studying vectors vs gradients or more generally from the study and generalization of the transformatiom properties of basis vectors vs the transformation properties of the coordinates of a vector expressed with respect to this basis. This is actually the way physicists are usually introduced to

understand cobases in a constructive way

(to borrow the terminology of the OP). A very clear and instructive exposition along these lines, emphasizing the phenomenological origin of this approach, can be found in chapters 2 and 3, of the classic text of B. Schutz, A first course in general relativity.

Edit: Maybe it would be important at this point, to note that the description of the transformation rules of displacement, velocity and acceleration vectors, under coordinate changes, are among the most fundamental and delicate problems of mechanics, if one is to build coherent definitions of these notions, to survive experiments ranging from subatomic to astronomical. They are pervading physical theories from the Galilean perception and Newtonian mechanics to relativity theory and continuoum mechanics and from Maxwell's electromagnetism to modern quantum field theories.
There are profound reasons for this: In physics, a system of coordinates (or a system of reference) is actually an observer. The study of the transformation rules of physical quantities, under coordinate or base changes, is not simply a theoretical exercise. It is actually a necessary step in the development of any physical theory, in the sense that it enables the seamless communication between different observers, that is between different experiments, which is crucial in accepting or rejecting any physical theory.
(On the other hand it should not be ignored that in the research level, modern theoretical physics strives for global and coordinate free descriptions. In my understanding this reflects the desire to pass from phenomenological descriptions to more fundamental theories).

P.S.: I am not sure i really agree with the use of the term "standard definition" in the OP. My first degree was on physics. I then joined the grad school on pure mathematics. I had a good "working understanding" of contravariant and covariant components, contractions, metric tensors, upper and lower indices etc. and i was quite comfortable in performing calculations with such objects. I still remember my astonishment when i first understood the abstract definition of the tensor product and its universal property, and notions such as dual spaces etc. I was striving -for weeks- to make the connections between the two definitions. When i finally managed to put things in some order in my head and to link what i already knew with what i learned in the grad school, i really felt something very important had happened to me: I felt i finally got to understand the ... "standard definition" ;)

(At the beginning of the grad school, i was desperately asking for help from my fellow students (most of them were coming from the undergrad math school) with the algebraic definition of the tensor product and the related notions (quotient spaces, universal properties etc). I still remember, and they probably also do, their surprise when they realized how easy were actual computations for me and when they started asking for my help with raising and lowering indices ... )

To conclude with a note on terminology: having read carefully through the various comments to the OP and to this post, i think that it might be sensible to speak of the present definition as the "phenomenological definition of a tensor", rather than the "standard definition", or the "physicist's definition" or the "pre-1930's mathematician's definition" or the "indices definition", just to collect a few of the terms that have been of use.

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    $\begingroup$ Konstantinos, however this lack of connection between the two definitions in textbooks creates a problem. Pedagogically it would be correct to discuss the different definitions and relations between them from the very begining. $\endgroup$ – Sergei Akbarov Mar 29 '19 at 21:36
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    $\begingroup$ I think it's a little unfair to call it the physicists' educational culture. Physicists care mostly about how things work and not the abstract concepts we use to explain why things work the way they do. So they focus mostly on the quantities that need to be calculated and the rules the calculations have to follow. They need to know how to use physical information to set up the inputs to a calculation and at the end interpret the outputs of the calculation as physical consequences. They have their own plausibiity arguments for deriving the right rules of calculations from physical principles. $\endgroup$ – Deane Yang Mar 29 '19 at 22:20
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    $\begingroup$ Historically this "physicists definition" was probably also mathematicians definition up to ~1930. So I prefer to think of it as "pre 1930 vs post 1930 mathematicians" instead of "physicists vs. mathematicians". $\endgroup$ – Michael Bächtold Mar 30 '19 at 10:12
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    $\begingroup$ @MichaelBächtold you are correct that mathematicians also used the ugly coordinate-dependent definition of tensors until the 1930s. And it's not true that all of physics is plagued with the "physicist's definition" of tensors: current textbooks on general relativity define tensors close to how the mathematicians do. See my discussion of tensor products in physics in the last section of kconrad.math.uconn.edu/blurbs/linmultialg/tensorprod.pdf. $\endgroup$ – KConrad Mar 30 '19 at 17:11
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    $\begingroup$ Perhaps a useful example to consider: Electric and magnetic fields transform in a subtle way under a Lorentz transformation. One doesn't really know a priori how they'll transform; one must rely on experimental observation (phenomenology). In some sense, the "computational" or "physicist" version of the transformation law is the true starting point, since it's what nature presents us with phenomenologically. Of course we then develop our basis-independent theoretical account to explain things. But if you're experimentally-minded then the computational version is closer to actual observations. $\endgroup$ – Timothy Chow Mar 31 '19 at 21:55
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To define tensors as they are used in physics with the clarity and rigour that is used in mathematics, we would have to develop the notions of:

  • duals of vector spaces
  • manifolds
  • vector bundles
  • the tangent bundle
  • tensors
  • and finally, tensor bundles

This is quite a bit of work and it's not surprising then that physicists eschew all this and choose to stick to their traditional definition and which most likely helped motivate the discovery of the above notions.

Still, given the clarity it brings to the subject I think that this is a pity. Even more so when one realises how omnipresent the bundle picture is in gauge theories like Yang-Mills. However, to bring all this to the attention of physicists would mean reforming the physics curricula and this itself is no small bit of work.

Personally, I can't have been the first person to have noticed that a contravariant and covariant vector is simply not a vector, either in its axiomatic or geometric definition. Even without chasing the full curriculum reform that the subject deserves, it might be worth pointing out where these new concepts fit in with the traditional definition.

I also think it's worth pointing out that the categorical definition of a tensor makes it quite clear we don't need to think of tensors as multilinear functions - we can define them directly. This merely underlines that there are a number of mathematical technologies that have been teased out of the traditional physical picture.

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I'm not able to enter comments with the phone that I have at the moment. So I'm answering comments here.

@Bachtold: The geometric picture of vectors that is usually found in most classical mechanics texts, and in the description of a vector that Einstein described in his semi-popular book, The Evolution of Physics. And the axiomatic definition that is found in most introductory books in linear algebra. And if my high-school curricula in a comprehensive school is any guide, in most schools.

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    $\begingroup$ Which definition of vector do you have in mind, when you conclude that contravariant and covariant vectors are not vectors? $\endgroup$ – Michael Bächtold Aug 25 at 14:43
  • $\begingroup$ Tensors can be added and multiplied with numbers, and these operations satisfy the usual axioms of a vector space, hence tensors are vectors. $\endgroup$ – Michael Bächtold Aug 27 at 8:46
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There are not just two, but at least 3 different ways to approach tensor products of vector bundles, each tailored to a different audience.

Intermediate between the "physicists' definition" mentioned in the question and the "definition in terms of tensor products of vector bundles" lies the "mathematician's pedagogical definition" which I remember encountering in do Carmo's textbook on Riemannian geometry (and I have the impression that it's widely used in introductory textbooks). Basically, this is just the definition in terms of tensor products in disguise, owing to the fact that textbook writers in Riemannian geometry don't want to assume that a student is familiar with tensor products. It seems that the author thinks in terms of tensor products, but avoids using the words "tensor product". In practice, this means they phrase everything in terms of the universal property of the tensor product, defining tensor products of (certain) vector bundles to be certain spaces of multilinear maps.

The category theorist in me is pleased that such a "hack" is made possible by thinking in terms of universal properties. The pedagogist in me is convinced that any math student who is exposed to vector spaces (an agreed-upon prerequisite for Riemannian geometry) should be exposed to tensor products. In an ideal world where this is the case, one could easily motivate the general definition of tensor product of vector bundles in an introductory Riemannian geometry course. Reasoning in terms of universal properties would still be useful, and would hopefully seem less mysterious to students.

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  • $\begingroup$ In geometric analysis, it is helpful to think somewhere between the fully-coordinate based approach of physics and the universal property approach of algebraists. While the latter is conceptually simpler in terms of defining what a tensor is in the first place, if you want to really prove analytic inequalities about say, positive Ricci curvature, its likely that you are going to have to deal with the former. $\endgroup$ – Gabe K Aug 25 at 18:15
  • $\begingroup$ As such, I suspect that many Riemannian geometers are really using this perspective when they introduce tensors to students. The goal is to compute curvature tensors and use their properties without getting too much into intricacies or working in full generality. $\endgroup$ – Gabe K Aug 25 at 18:29

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